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Abstract
The dilemma between tunable range of a lens power and an aperture size in gradient-index (GRIN) type of liquid crystal (LC) lenses is well known due to the limitation of birefringence of LC martials and the nature of soft matters. In order to overcome the dilemma of the power law, a multi-layered LC lens was previously proposed by us. However, the aperture size of GRIN LC lenses is still difficult to exceed 10 mm. In this paper, we proposed an electrically tunable GRIN LC lens via nematic liquid crystals with a method of spatially extended phase distribution. A GRIN LC lens with an aperture size of 20 mm is achieved. The proposed GRIN LC lens consists of two LC elements modulating wavefronts at different regions of the lens aperture extending to 20 mm. The lens power of the GRIN LC lens is continuously tunable, while the LC lens can function as a positive lens, a negative lens, and a bifocal lens. The proposed GRIN LC lens not only overcomes the dilemma of the power law but also provides a more practical approach that could benefit the ophthalmic applications and augmented reality systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Liquid crystal (LC) lenses attract lots interests in optical engineering because of electrically tunable focusing capability. The optical mechanism of LC lenses is based on a modulation of light speed resulting from LC molecular orientations harnessed by applied electric fields [1−3]. Many applications of LC lenses are developing so far, such as contact lenses [4–5], eyeglass [6−8], and vision correction elements in augmented reality systems and virtual reality system [9−11]. Typically, LC spatial light modulators modulate phase retardations of light pixel by pixel, but the spatial distribution of the phase retardation is step-like (digital phase retardation) but not continuous [12−14]. This leads to extra aberration in the optical systems. To obtain a continuous distribution of a phase retardation (analog phase retardation), four main types of LC lenses have been developed: curved lenses, gradient-index (GRIN) lenses, the mixed type, polarization-switching-type LC lenses [2]. However, achieving larger aperture (>10mm) is always a challenge, and then the development of LC lenses especially in ophthalmic applications (e.g., 20mm for eyeglasses) is restricted due to limited birefringence of LC martials (<0.8) [15] and the nature of soft matters. To enlarge the aperture size of the LC lenses, researchers are developing LC lenses on a basis of three types: Pancharatnam-Berry phase type [16−18], Fresnel type [2–3,19−21], and GRIN type [2,22−24]. The Pancharatnam-Berry phase type is so-called a waveplate lens with lots of sub-waveplates. Each sub-waveplate is half-wave plate (i.e. the phase retardation is half-wave) and the fast/slow axis of each sub-waveplate is spatially arranged. A half-wave plate helps to convert one circularly-polarized light to another circularly-polarized light with an extra optical phase related to the angle of the fast axis [17]. The waveplate lens is thin and easy to achieve large aperture size, but the continuity of the spatial distribution in the optical phase depends on how small each sub-waveplate is during fabrication process. It is still step-like spatial distribution of the phase retardation if the resolution is not enough. In addition, the waveplate lens has two discrete lens powers and the lens power (an inverse of focal length) is not continuously adjustable. The Fresnel type of LC lenses could achieve large aperture size by combining refraction and diffraction [20–21,25–26]. Such a Fresnel type of LC lens is so-called kinoform lens or blazed-phase Fresnel zone plate. Researchers usually divide a phase zone of 2π radians into many step-like subzones to mimic the continuous phase profile of a lens [2–3,20–21]. Light efficiency of Fresnel lenses is limited to the number of subzones. For example, light efficiency of two sub-steps and eight sub-steps are 40.5% and 95.0%. The fabrication limits the area of outer zones and complexity of electrodes design is also challenging. In addition, the edges of rings reduce the image quality. The spatial distribution of phase retardation of the Fresnel type of LC lenses is still step-like, not continuous. Moreover, the focal lengths of the Fresnel type of LC lenses are not continuously tunable with applied voltages [2–3,20–21]. Intrinsically, Fresnel LC lenses with optical phase discontinuity are general not suitable for optical systems requiring good image quality and an ability for aberration correction. GRIN type of LC lenses features a gradient distribution of refractive index owing to the spatially inhomogeneous electric field applying to LC layers, where the lens power of a GRIN LC lens is also continuously tunable [2,22−24]. However, the development of GRIN LC lenses is limited by the power law [2,7–8] - the dilemma between tunable range of the lens power and the aperture size especially when the thickness of a LC layer and the birefringence of LC are limited. In order to overcome the dilemma of the power law, we demonstrated a multi-layered LC lens with single set of electrodes, which is able to augment the optical phase to keep the same lens power when the aperture size increases [7–8]. When the aperture size is up to 20 mm, it requires at least 8 layers for a tunable range of 4.8 diopter. In practical, the transmittance of 8-layered LC structure could be low, and it is not easy to manufacture for mass production. It motivates us to keep challenging GRIN LC lenses with large aperture size. In this paper, we proposed an electrically tunable GRIN LC lens via nematic liquid crystals with a method of spatially extended phase distribution. A GRIN LC lens with an aperture size of 20 mm is achieved. The proposed GRIN LC lens consists of two LC elements. One of the LC element is responsible for modulating the optical phase in a diameter of 10 mm. The other LC element modulates the optical phase in the diameter between 10 mm and 20 mm. Two LC elements could be driven independently for different spatial locations. By manipulating applied voltages and frequencies of two LC elements, the lens power of the GRIN LC lens is continuously tunable and the modulated wavefront is also adjustable which could function as a positive lens, a negative lens, or bifocal lens. Notably, the bifocal lens is not easy to achieve in the multi-layered LC lens with a single set of electrodes [8]. The merits of the proposed method are not only the flexibility to increase the aperture size by adding LC elements without re-designing or re-fabricating the whole LC structure but also the capability of wavefront aberration because each LC layer could be addressed independently. Compared to the multi-layered LC lens we proposed previously [8], this method could reduce the number of LC layers while overcome the dilemma of the power law of a GRIN LC lens. The transmittance loss of the proposed method may come from Fresnel reflection between LC elements. This issue can be solved by coating anti-reflection layers, which is not easy to implement in multi-layered LC structure with a single set of driving electrodes [8]. This study provides a more practical approach which could benefits applications, such as ophthalmic lenses and augmented reality systems.
2. Operating principles and sample preparation
To proof-of-concept, the structure of the proposed LC lens is illustrated in Fig. 1(a)–1(b). We fabricated two LC elements and then stacked them together. The first LC element was composed by two sheets of an indium tin oxide (ITO) layer and a hole-patterned ITO layer to generate inhomogeneous electric fields. The radius (R1) of a hole-patterned ITO layer is 5 mm (i.e., the aperture size of 2R1 = 10 mm). The hole-patterned ITO was coated on a glass substrate with thickness of 0.4 mm. A sheet ITO layer and the hole-patterned ITO layer were separated by a spin-coated high resistive layer (HRL) and an insulating layer (NOA 81, Norland) with thickness of 35 µm. HRL is made of conductive polymer (Dongjin Semichem Conducting Polymer DJCP series, Dongjin Semichem Co., Ltd) whose sheet resistance was controlled as ∼3MΩ/sq and the thickness was < 100 nm. Two mechanically buffered alignment layers (polyvinyl alcohopolyvinyl; homogenous alignment) were coated on a sheet ITO glass substrate and the other side of the glass substrate with the hole-patterned electrode in order to align the LC mixture (nematic LC, Merck, MLC-2172, Δn = 0.29 at λ = 589.3 nm), where the thickness of the LC layer (i.e., cell gap) was 100 µm controlled by a Mylar film. The 100-µm thick LC layer provide ∼49 waves of phase retardation. The second LC element is similar to the first one. The differences are the thickness of a LC layer ∼50 µm and the radius (R2) of the hole-patterned electrode was 10 mm (aperture size 2R2 = 20 mm); The hole-patterned ITO electrode was coated on a 0.7 mm-thick glass substrate. The total aperture size of two LC elements was 20 mm.
Fig. 1. (a) Illustration of the structure of the proposed LC lens. (b) is the conceptual refractive index as a function of radius for the top LC element or the first LC element in (a). (c) is the conceptual refractive index as a function of radius for the bottom LC element or the second LC element in (a). (d) is sum of (b) and (c).
The optical phase transfer function of the proposed LC lens (Fig. 1(a)) is the sum of two LC elements. Since two LC elements are based on the principle of GRIN lens, which features spatially continuous refractive index for all positions, the proposed LC lens has no optical phase discontinuity in principle. It is feasible to further enlarge the effective aperture size of the LC lens by stacking more LC elements, where the general expression of phase transfer function is
3. Experiments and discussions
After the fabrication of the sample, a Shack-Hartmann wavefront sensor (WFS150-7AR, Thorlabs) was used to measure wavefronts of the sample and the detail measurement was reported previously [10–11]. In short, the light source in the wavefront measurement system consists of a laser (a diode pumped solid state laser, λ = 532 nm), a single mode fiber to form a point source, and a lens with a focal length of 75.6 cm for converting the point source into plane waves. The collimated plane waves propagate through a linear polarizer and then go to the sample (i.e., LC lens). After light propagates to the LC lens, the output wavefront from the LC lens is relayed to the wavefront sensor by a pair of solid lenses as relay optics. The paired solid lenses are set as a confocal configuration, and the LC lens is placed in the front focal plane of the first solid lens while the wavefront sensor locates at the back focal plane of the second solid lens. For the measuring area of 10 mm (aperture size), the focal lengths of two solid lenses used are 16.7 cm and 7.1 cm, respectively. For the aperture size of 20 mm, the focal lengths of the paired lenses we used are 33.3 cm and 7.1 cm, respectively. In the measurement, the wavefront sensor provides us the fitting data in the form of 10th order Zernike coefficients (${c_i}$, $i$=0-65). The wavefront (W) is expressed by Zernike polynomials as: $W = \sum\limits_i {{c_i} \cdot {Z_i}}$, where Z is Zernike polynomials [27]. The lens power (P) can also be expressed as: $P ={-} 4\sqrt 3 \cdot {c_4}/r_0^2$, where ${r_0}$ is radius of the aperture size [27]. For the first LC element (the aperture size 2R1 = 10 mm), the measured lens power as the function of a voltage pair of (V1, V2) at different AC (alternating current) frequencies is shown in Fig. 2(a). The measured lens power ranges from −2.21 Diopter (D) to + 2.26 D (i.e., 4.5 D in total) when the applied voltage is less than 20 Vrms at the frequency ranging from 200 Hz to 2600 Hz. Then, we select some voltage conditions for further experiments based on the criteria of paraboloidal wavefront (i.e., less spherical aberration). The cross-sections of wavefronts at selected voltage conditions are shown in Fig. 2(b), where the lens power could be from −1.9 D to + 1.9 D (i.e., 3.8 D in total). Here, we checked the properties of the first LC element with the aperture size of 10 mm. The applied voltage could be reduced to less than 10 Vrms for practical applications if a thinner glass substrate between the hole-patterned electrode and the LC layer is implemented. The operating frequency could be adjusted for the driving circuit by changing the sheet resistance of the high-resistive layer [2,28]
Fig. 2. (a) The measured lens power as a function of an applied voltage pair (V1, V2) at different AC frequency. (b) Wavefront cross-sections of the first LC element as a function of y-pupil coordinate with an aperture size of 10 mm.
The wavefronts of two LC elements and the LC lens (attached by two LC elements) were measured for the aperture size of 20 mm (i.e. 2R2 in Fig. 1(a)). To test the LC lens, we first tested the region of the first LC element at $|r |$ < R1 (Fig. 3(a)) and then tested the region of the second LC element at R2 > $|r |$ > R1 (Fig. 3(b)). Figure 3(a) plotted wavefront cross-sections as the function of y-pupil coordinate of the LC lens when we applied voltage pairs of (V1, V2) at different frequencies to the first LC element; meanwhile, the second LC element was off. In Fig. 3(a), the lens power could be switched from 0.0 D to 1.9 D at $|r |$ < 5 mm (or the aperture size is 10mm) when the lens power of the second LC element is 0.0 D. The LC lens functions as a positive lens at $|r |$ < 5 mm. Figure 3(b) shown wavefront cross-section as the function of y-pupil coordinate of the LC lens when we applied different voltage triplets of (V3, V4, V5) at different frequencies to the second LC element for effective lens power at R2 > $|r |$ > R1 ranging from −0.3 D to + 0.3 D; meanwhile, the first LC element was off (i.e., zero lens power). In Fig. 3(b), the wavefront is almost flat as $|r |$ < R1, but the wavefront at r = 10mm (or −10mm) could be larger (blue line) or smaller (red line) than the wavefront at r = 5mm (or −5mm). This indicates the optical path at the center (r = 0) is longer (red line) or shorter (blue line) than the peripheral region ($|r |$=10 mm). The second LC lens could help to adjust the wavefront of the LC lens at R2 > $|r |$ > R1. From Figs. 3(a) and (b), the LC lens could be operated as a positive lens as (V1, V2) and frequencies for the first LC element change while (V3, V4, V5) = (40 Vrms, 0 Vrms, 40 Vrms) at 200 Hz for the second LC element, as shown in Fig. 3(c). We could also estimate the lens power by means of the relation: the effective lens power (P) equals to $2 \cdot \delta n \cdot d/{r^2}$, where $\delta n \cdot d$ is wavefront from r = 0 to r = 10 mm [2]. In Fig. 3(c), the effective lens power across entire aperture (2R2) ranging from 0.4 D to 0.7 D. One can see that the flat wavefront in Fig. 3(b) (i.e. red line in Fig. 3(b) at $|r |$ < R1) helps to lift up the wavefront of the LC lens at $|r |$ < R1 in Fig. 3(c). The second LC element helps to add a bias wavefront at $|r |$ < R1 when the first LC element modulates the wavefront of the LC lens. At R2 > $|r |$ > R1, the second LC element is in charge of wavefront modulation of the LC lens. Similarly, Fig. 3(d) shows that the LC lens (two LC elements) could be operated as a negative lens with different (V1, V2) and frequencies for the first LC element while (V3, V4, V5) = (0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz for the second LC element. The effective lens power across entire aperture (2R2) changes from −0.3 D to −0.6 D. No matter in Fig. 3(c) or Fig. 3(d), the measured wavefronts display continuously convergent or divergent wavefronts within aperture size of 20 mm. Thus, by manipulating two LC elements, we are able to realize GRIN LC lens with the aperture size of 20 mm.
Fig. 3. Wavefront cross-section as the function of y-pupil coordinate for the LC lens. The measured aperture size is 20 mm. (a) The first LC element is applied with selected voltage pairs (V1, V2) at different frequencies and the voltage triplet of the second LC element is fixed as (0 Vrms, 0 Vrms, 0 Vrms). (b) The second LC element is applied with different voltage triplets of (V3, V4, V5) and the voltage pair of the first LC element is fixed as (0 Vrms, 0 Vrms). (c) The LC lens (two LC elements) is operated as a positive lens as (V1, V2) and frequencies for the first LC element change. (V3, V4, V5)=(40 Vrms, 0 Vrms, 40 Vrms) at 200 Hz for the second LC element. (d) The LC lens (two LC elements) is operated as a negative lens when V1 > V2 for the first LC element while (V3, V4, V5)=(0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz for the second LC element.
To visualize and exam the phase profiles of the LC lens, the LC lens was placed between two crossed polarizers. The alignment directions of two LC elements were parallel to each other, and they are 45 degrees with respect to the transmissive axis of one polarizer. A Helium-Neon laser (λ = 543 nm) was used as the light source and the phase profiles was captured by a camera, as shown in Figs. 4(a) –4(f). In Figs. 4(a)–4(f), the concentric rings represent the distribution of phase retardation of the LC lens. The adjacent intensity peaks (or two adjacent rings) means difference of phase retardation of 2π radians. In Fig. 4(a), when the first LC element is applied with voltage of (10 Vrms, 20 Vrms) at 1000 Hz (−0.6 D lens power) while the second LC element is off, the phase retardation mainly distributed in the center area with 10 mm in diameter. When the first LC element is off and the second LC element is operated with a voltage triplet (0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz for negative lens power (see Fig. 4(b)), the center area has uniform phase retardation while the area at 10 mm > $|r |$ > 5 mm exhibits a concentric pattern. The non-perfect area in Fig. 4(b) is caused by the ITO bus line for connecting between the inner and outer ITO patterns as shown in Fig. 1(a); this could be prevented using addressable electrode design (e.g., vias) [1]. In addition, the boundary between the two LC layers can be smooth when the focal lengths of two LC layers are identical or have a gradient distribution. The non-smooth boundary between the two layers could be improved with well alignment of optical axes of two LC layers. From the results in Figs. 4(a) and 4(b), two LC elements modulate the wavefront at different spatial locations with gradually various refractive index. When two LC elements are operated together for a negative lens power, the effective aperture size reaches to 20 mm, as shown in Fig. 4(c). It is clear that, the LC lens features a spatially continues phase profile. Similarly, when two LC elements are operated for a positive lens power (see Figs. 4(d)–4(f)), the phase profile is also spatially extended to the aperture size of 20 mm.
Fig. 4. Phase profiles of the LC lens. (a) The first LC element is operated as (V1, V2) = (10 Vrms, 20 Vrms) at 1000 Hz and the second LC element is off. (b) The first LC element is off when the second LC element is operated as (V3, V4, V5) = (0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz. (c) The first LC element is operated as (V1, V2) = (10 Vrms, 20 Vrms) at 1000 Hz and the second LC element is operated as (V3, V4, V5) = (0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz. (d) The first LC element is operated as (V1, V2) = (20 Vrms, 0 Vrms) at 900 Hz and the second LC element is off. (e) The first LC element is off when the second LC element is operated as (V3, V4, V5) = (40 Vrms, 0 Vrms, 40 Vrms) at 200 Hz. (f) The first LC element is operated as (V1, V2) = (20 Vrms, 0 Vrms) at 900 Hz and the second LC element is operated as (V3, V4, V5) = (40 Vrms, 0 Vrms, 40 Vrms) at 200Hz. The LC lens in (c) and (f) are a negative lens (∼ −0.3 D) and a positive lens (∼ +0.4 D) with the aperture size of 20 mm. The P and A arrows mean transmissive axes of polarizer and analyzer. The R arrow means rubbing directions of two LC elements.
The image performance of the LC lens was taken by a camera (Canon, EOS 760D) under ambient white light when the LC lens was located at 40 cm away from the camera (s2 = 40 cm), as shown in Fig. 5. A printed paper with a text of “NCTU” was placed at s1 = 18 cm away from the LC lens as a target (i.e. s1+s2 = 58 cm). An iris with a diameter of 20 cm attached to the LC lens for shielding from the ambient light. When the two LC elements are off, the captured image is shown in Fig. 5(a). The LC lens composed of two regions ($|r |$<5 mm and 10mm > $|r |$ > 5mm) and two regions could have the same lens power or not. As a result, we adjust the lens module of the camera in order to see either clear images at center region of $|r |$<5 mm (Figs. 5(b) –5(e)) or at the outer region of 10mm > $|r |$ > 5mm (Figs. 5(f)–5(i)). Thereafter, we applied electric fields to the second LC element with applied voltage triplet of (40 Vrms, 0 Vrms, 40 Vrms) at 200 Hz and then we changed the lens power of the first LC element from 0, +0.6 D, +1.2 D and + 1.9 D. The lens power of the first LC element and the corresponding voltage pairs of (V1, V2) and frequencies are list in the caption of Fig. 5. From Figs. 5(b)–5(e), the central images are magnified when the lens power of the first LC element increases. The images at the outer region (10mm > $|r |$ > 5mm) are blurred in Figs. 5(b)–5(e) and this indicates the difference lens power between the central region and the outer region. Next, we re-adjusted the lens module of the camera in order to see clear image at the outer region of 10mm > $|r |$ > 5mm (Figs. 5(f)–5(i)). The voltage conditions are identical to Figs. 5(b)–5(e), but the camera sees the images clearly at the outer region. As we can see in Figs. 5(f)–5(i), the images at the outer region are similar, but the central images become more blurred in Fig. 5(i). This indicates the difference of lens powers between the central region and the outer region increases which is agreeable with the results in Fig. 3(c). When such a difference of lens powers between the central region and the outer region is larger, the photos exhibit the bifocal nature of the LC lens. Two movie files of the process of adjusting the camera can be found in Supplementary Material: one is for Figs. 5(b) and 5(f) (see Visualization 1); another one is for Figs. 5(e) and 5(i) (see Visualization 2).
Fig. 5. Image performances of the LC lens when (a) two LC elements are at the voltage-off state. (b) to (i): the first LC lens is operated with a various lens power when the second LC element is applied with a voltage triplet of (40 Vrms, 0 Vrms, 40 Vrms) at 200 Hz. (b) to (e) are image performances when the camera was focused in the center region ($|r |$ < 5 mm) and (f) to(i) are image performances when the camera was focused in the out region (10 mm > $|r |$ > 5 mm). (b) and (f): the lens power of the first LC element was zero (see Visualization 1). (c) and (g): the lens power of the first LC element was 0.6 D as the applied voltage pair of (20 Vrms, 0 Vrms) at 900 Hz. (d) and (h): the lens power of the first LC element was 1.2 D as the applied voltage pair of (20 Vrms, 10 Vrms) at 2600 Hz. (e) and (i): the lens power of the first LC element was 1.9 D as the applied voltage pair of (20 Vrms, 5 Vrms) at 2200 Hz (see Visualization 2).
The LC lens we demonstrated above could be a positive lens or negative lens. Furthermore, we could also design different and electrically tunable bifocal lens based on the same LC lens structure or concept by adjusting each LC element with parameters, such as voltages, frequencies, thickness of the LC layers, and birefringence of LC materials (Δn) based on the needs of wavefront modulation. Here we discuss an example of design. For a given focal length of the first LC element of f1, the phase transfer function of the first LC element is:
Fig. 6. Wavefront cross-sections as the function of y-pupil coordinate when the LC lens is operated as an electrically tunable bifocal LC lens. The second LC element is operated with (0 Vrms, 40 Vrms, 40 Vrms) at 50 Hz while changing the voltage pairs and frequencies of the first LC element.
4. Summary
We present a LC lens based on GRIN LC structures for enlarging the effective aperture up to 20 mm in diameter. The aperture size of the LC lens could be further extended with more LC optical elements, and the concept is easy to be adopted with other type of tunable lenses. The fabrication of LC samples could be simplified and practical for optical systems. The LC lens we proposed could feature different response time, tunable range of the lens power, and even functions of aberration correction at different spatial locations because each LC element could be addressed independently. The LC lens could be extended as a freeform optics in principle. Since each LC element is based on GRIN type of LC lenses, the proposed LC lens here is spatially continuous without optical phase discontinuity. The LC lens could be a tunable bifocal lens or even a freeform optical element. The proposed concept here could overcome the power law [2,7–8] of the GRIN LC lens, a dilemma between the aperture size and the tunable lens power of GRIN LC lenses. The impact of this study is in applications of tunable ophthalmic lenses and geometrical optical system design [6−11].
Funding
Ministry of Science and Technology, Taiwan (107-2112-M-009-019-MY3); Coretronic Corporation.
Acknowledgments
The authors are indebted to Dongjin Semichem Co., Ltd for providing the conductive polymer (Dongjin Semichem Conducting Polymer DJCP series).
Disclosures
YJW: Coretronic Corporation (F), HAH: Coretronic Corporation (F), YHL: Coretronic Corporation (F).
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